Medical imaging systems such as computed tomography (CT) and magnetic resonance imaging (MRI) employ computer reconstruction techniques to convert acquired data into tomographic or "slice" images of a patient.
In CT, the acquired data consists of radiographic projections obtained with a fan beam of x-rays at regular angular intervals about the patient within the plane of the slice. A Fourier transformation of each projection provides a line of the Fourier transform of the desired slice image along a line of diameter about the center of rotation of the projections. A set of projections at different angles, therefore, provides a set of different lines of Fourier data arranged like spokes in a wheel. This Fourier data is converted to a rectangular or Cartesian format and the two-dimensional inverse Fourier transform is taken to produce the tomographic image.
In MRI, the acquired data is a sampled nuclear magnetic resonant (NMR) signal received from the patient after stimulation by a radio frequency (RF) electromagnetic field and during the application of one or more magnetic gradient fields. The gradient fields are produced by amplifiers driving conductive coils so the gradient fields may be varied in a "sequence" during the imaging. Multiple NMR signal acquired during the sequence, produce a field of data within a Fourier "k-space". This field of data is operated on by the two dimensional Fourier transform to produce the tomographic image. In one gradient sequence that reduces the peak power required of the gradient amplifiers, each acquisition of an NMR signal produces data along a spiral path through the k-space. As with CT, it is typical to convert this data to a Cartesian format to process and display it.
In both CT and MRI, each acquired data point is associated with an elemental measurement volume (voxel), either in the patient or in the Fourier space of the transformed data. As a result of the radial nature of the acquisition, the voxels are sectors of cylindrical annuluses arranged about a common center within a slice plane, their exact dimensions being determined by the resolution of the system. Normally, the thickness of the voxels (measured normal to the image plane) is constant and thin so that the voxels may be treated practically as areas rather than volumes. Voxels of this type will be termed "polar voxels" and the associated delta will be termed "polar data".
As noted above, accurate and efficient processing of a tomographic image requires that the polar data be converted to data values associated with rectangular voxels arranged in lines. These voxels will be termed "Cartesian voxels" and their associated data will be termed "Cartesian data". Cartesian data is easier to process and most commercial display devices, such as CRTs, enforce the use of data in a Cartesian format as a result of their scanning pattern.
Ideally, the conversion between polar data and Cartesian data accurately reflects the relative size, shape, orientation and position of the associated polar voxels and Cartesian voxels. Thus, a simple interpolation between polar data and closest Cartesian data based on the centers of the voxels is unacceptable.
Accurate conversion reflecting the relative size, shape, orientation and position of the associated polar voxel and Cartesian voxel can be difficult. There is no simple relationship between the position and relative orientation of polar voxels and Cartesian voxels. This is compounded by the fact that the shape and area of the polar voxels will generally change depending on their distance from the center of the polar reference frame.
The intractability of this conversion process has led to the development of an oversampling technique where the polar voxels are superimposed on an extremely fine Cartesian grid and each of the Cartesian voxels is cataloged as being inside or outside a given polar voxel. After a large number of such points are cataloged, the number of Cartesian voxels in each polar voxel is used to establish a conversion weight. Although this technique can provide an arbitrarily high degree of accuracy, it is extremely inefficient and time consuming.
A table incorporating the conversion weights may be used for multiple conversions provided the relative size, number and positioning of the polar and Cartesian voxels is not changed. Any change in the relative size, number or positioning of the polar and rectangular coordinates, however, requires a completely new calculation of this table. There is no simple operation on the conversion weights that will produce a new table for such changes in the structure of the rectangular or polar voxels. Thus the need for a computationally efficient means for converting polar data to Cartesian data or generating a conversion table is needed.